I wanted to know why the contactor is not closing to start the compressor and outdoor fan motor, my indoor fan is coming on.

Because it needs replaced.

We need more info if you can give it. Is it a heatpump or just a/c, how old is it 10, 15, 20 years old? Some of the systems use pressure switches and time delay relays that 24v has to pass through before the contactor.

If the contactor is not closing,, are you sure the low voltage (24volts ) is going to contactor to turn it on or kick it in,, can you push it in manually with a srewwdriver and allow it to kick back out by itself,, or test with a meter for 24 volt power

We need more info if you can give it. Is it a heatpump or just a/c, how old is it 10, 15, 20 years old? Some of the systems use pressure switches and time delay relays that 24v has to pass thru before the contactor.

I can push it in with a screwdriver and compressor and fan run

Is it a heat pump? Check the voltage on the (normally) yellow wire. It should be 24 VAC. If it is not approximately that value, check to see if you have a float switch. If the float switch is activated or defective, the outside unit on a split system will not kick in.